Explicit method calculates yn+1 when we already know yn. In other words, yn+1 is explicitly shown as a function of yn: yn+1 = f(yn).

Explicit Euler method:

Implicit method calculates yn+1 from an equation involving both yn+1 and yn. It considers yn+1 as an unknown variable.

Implicit method requires extra computation and harder. It is used to solve stiff equations.

Implicit Euler method: . It’s called implicit because it requires information at xn+1.

Stiff equations

Stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable (for a change of independent variable x will produce a very high value of dependent variable y) unless the step size is taken to be extremely small (very small change of x).

When an explicit method is used to solve stiff equations, it requires impractically small time steps (∆t). Hence, it needs longer computation time.